\contentsline {chapter}{\numberline {1}\color {.} Preface: why this book and how to read it}{7}{chapter.1}
\contentsline {chapter}{\numberline {2}What is a tracking code: PTC used as example}{11}{chapter.2}
\contentsline {section}{\numberline {2.1}Why do we use ``lenses'' and not time tracking?}{11}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Concept of a Beam Line}{11}{subsection.2.1.1}
\contentsline {subsection}{\numberline {2.1.2}Patches: Translations and Rotations}{11}{subsection.2.1.2}
\contentsline {section}{\numberline {2.2}Explicit Symplectic Integration with examples}{11}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}Quadrupole}{11}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Dipoles}{11}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}Rectangular Bends}{11}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}Pill Box Cavities}{11}{subsection.2.2.4}
\contentsline {subsection}{\numberline {2.2.5}Non-Symplectic Elements}{11}{subsection.2.2.5}
\contentsline {chapter}{\numberline {3}The ray and its extensions}{13}{chapter.3}
\contentsline {section}{\numberline {3.1}What can we compute besides the ray itself?}{13}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}Extend the ray by adding physics: spin and stochastic envelopes}{13}{subsection.3.1.1}
\contentsline {subsection}{\numberline {3.1.2}Extend the ray by adding useful mathematics: Taylor series and maps for beam lines}{13}{subsection.3.1.2}
\contentsline {section}{\numberline {3.2}FPP: A Polymorphic Taylor Type }{13}{section.3.2}
\contentsline {subsection}{\numberline {3.2.1} Maps for the Orbital Dynamics}{13}{subsection.3.2.1}
\contentsline {subsection}{\numberline {3.2.2} Maps for the Spin Dynamics}{13}{subsection.3.2.2}
\contentsline {subsection}{\numberline {3.2.3} Maps with Parameter Dependence}{13}{subsection.3.2.3}
\contentsline {chapter}{\numberline {4}Analysis in FPP/PTC: Normal Form for Invariants}{15}{chapter.4}
\contentsline {section}{\numberline {4.1}\color {.} A bit of theory to show the usefulness of normal forms: invariants}{15}{section.4.1}
\contentsline {section}{\numberline {4.2}\color {.}Courant-Snyder: The quadratic invariants in one or more degrees of freedom}{16}{section.4.2}
\contentsline {section}{\numberline {4.3} \color {.}The nonlinear invariants in one or several degrees of freedom}{17}{section.4.3}
\contentsline {section}{\numberline {4.4} \color {.} The tunes: in glorious generality (well spin excluded here)}{18}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}\color {.}The case of a (damped) nonlinear oscillator}{19}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}\color {.}The case of nonlinear oscillator with a constant energy in the longitudinal plane}{23}{subsection.4.4.2}
\contentsline {section}{\numberline {4.5}\color {.} The spin invariant beyond linear: the vector $\mathaccentV {vec}17E{n}(\mathaccentV {vec}17E{z})$ }{23}{section.4.5}
\contentsline {subsection}{\numberline {4.5.1}\color {.} Normal form using the real FPP }{24}{subsection.4.5.1}
\contentsline {subsection}{\numberline {4.5.2}\color {.} Comparison with Stroboscopic Tracking}{27}{subsection.4.5.2}
\contentsline {chapter}{\numberline {5}Analysis in FPP/PTC: Normal Form for Averages}{29}{chapter.5}
\contentsline {section}{\numberline {5.1}\color {.} A bit of theory to show the usefulness of normal forms}{29}{section.5.1}
\contentsline {section}{\numberline {5.2}\color {.} A bit of FPP implementation of the theory}{32}{section.5.2}
\contentsline {section}{\numberline {5.3}\color {.} $\left \delimiter "426830A {{z}_{i}}\right \delimiter "526930B $: various dispersions }{33}{section.5.3}
\contentsline {subsection}{\numberline {5.3.1}\color {.} Amplitude dependent dispersions: general }{33}{subsection.5.3.1}
\contentsline {subsection}{\numberline {5.3.2}\color {.} Linear and Nonlinear dispersions: quick and dirty }{35}{subsection.5.3.2}
\contentsline {section}{\numberline {5.4}\color {.} Time or path length average: $\left \delimiter "426830A {{z}_{6}}\right \delimiter "526930B $ (PTC's longitudinal variable) }{36}{section.5.4}
\contentsline {subsection}{\numberline {5.4.1}\color {.} Time average: Momemtum Compaction linear and nonlinear }{36}{subsection.5.4.1}
\contentsline {subsection}{\numberline {5.4.2}\color {.} 3-degrees-of-freedom Jordan Normal Form: Chromaticities!}{39}{subsection.5.4.2}
\contentsline {section}{\numberline {5.5}$\left \delimiter "426830A {{z}_{i}{z}_{j}}\right \delimiter "526930B $ : Standard Lattice Functions }{40}{section.5.5}
\contentsline {subsection}{\numberline {5.5.1}\color {.} General computation: amplitude dependent lattice functions }{40}{subsection.5.5.1}
\contentsline {subsection}{\numberline {5.5.2}\color {.}Lattice Function: quick and dirty }{42}{subsection.5.5.2}
\contentsline {subsection}{\numberline {5.5.3}Tracking a Spin Transformation: SLICK code }{43}{subsection.5.5.3}
\contentsline {subsection}{\numberline {5.5.4}Coefficients of the Invariants: duality with Moments }{46}{subsection.5.5.4}
\contentsline {subsection}{\numberline {5.5.5}Complete construction of the $\delta $ dependent matrix with the moments and the tunes }{46}{subsection.5.5.5}
\contentsline {chapter}{\numberline {6}\color {.} The Phase advance }{47}{chapter.6}
\contentsline {section}{\numberline {6.1}\color {.} General Theory: the effect of putting on a pair of glasses}{47}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}\color {.} Phase advance: the need for more maps}{48}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}\color {.} On the lack of uniqueness of the phase advance}{49}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}\color {.} Tracking $a(s)$ and the phase advance}{49}{subsection.6.1.3}
\contentsline {subsection}{\numberline {6.1.4}\color {.} Is there a special choice $a(s)$?}{50}{subsection.6.1.4}
\contentsline {section}{\numberline {6.2}\color {.}(Non)Linear Phase Advance: how to write a Twiss Loop}{52}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1} \color {.}The fast and dirty way}{52}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2} \color {.}The not-so-fast and dirty way to handle parameters}{54}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}\color {.}The more general (slow) way: linear and nonlinear}{55}{subsection.6.2.3}
\contentsline {chapter}{\numberline {7} \color {.} Tracking a stochastic quadratic envelope with radiation}{57}{chapter.7}
\contentsline {section}{\numberline {7.1} \color {.} Getting beam sizes with this exact linear theory}{57}{section.7.1}
\contentsline {section}{\numberline {7.2} \color {.} Getting the approximate Chao synchrotron integral from \ref {secsize}}{59}{section.7.2}
\contentsline {chapter}{\numberline {8}The one-resonance Normal Form }{63}{chapter.8}
\contentsline {section}{\numberline {8.1}The Trivial method of Dragt for 1-d resonance}{63}{section.8.1}
\contentsline {section}{\numberline {8.2}The method of Turchetti and Forest for higher dimensionality}{63}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}\color {.} Example of the Turchetti Theory in 1-d}{63}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2} A 2-d example}{64}{subsection.8.2.2}
\contentsline {section}{\numberline {8.3}Spin resonance: here the power of our methods outshines the rest!}{64}{section.8.3}
\contentsline {subsection}{\numberline {8.3.1}Example with an ordinary resonance:\nobreakspace {} ${\nu }_{x}+{\nu }_{{\rm s}{\rm p}{\rm i}{\rm n}}= 1$ }{64}{subsection.8.3.1}
\contentsline {subsection}{\numberline {8.3.2}Example with a magnet modulation resonance:\nobreakspace {} ${\nu }_{{\rm m}{\rm o}}+{\nu }_{{\rm s}{\rm p}{\rm i}{\rm n}}= 1$ }{64}{subsection.8.3.2}
\contentsline {chapter}{\numberline {9}Miscellaneous Useful Stuff in FPP}{65}{chapter.9}
\contentsline {section}{\numberline {9.1} Dragt-Finn Representation}{65}{section.9.1}
\contentsline {section}{\numberline {9.2} One-Lie Exponent representation}{65}{section.9.2}
\contentsline {section}{\numberline {9.3} Mixed generating functions and their practical use (Symplectic tracking and BPM reconstruction)}{65}{section.9.3}
\contentsline {section}{\numberline {9.4} Forcing $A$ in a particular form to all orders: gives phase advance}{65}{section.9.4}
\contentsline {section}{\numberline {9.5} Symplectify maps or checking the symplectic condition}{65}{section.9.5}
\contentsline {section}{\numberline {9.6} Inverting general maps}{65}{section.9.6}
\contentsline {section}{\numberline {9.7} Solving Maxwell Equation in the bend with TPSA }{65}{section.9.7}
\contentsline {chapter}{Appendices}{}{section*.2}
\contentsline {chapter}{Appendix \numberline {A}\color {.} Normal Form of the Tiny Package}{69}{Appendix.1.A}
\contentsline {section}{\numberline {A.1}\color {.} Normal Form in General}{69}{section.1.A.1}
\contentsline {section}{\numberline {A.2}\color {.} Some explanations of the actual code}{69}{section.1.A.2}
\contentsline {subsection}{\numberline {A.2.1}\color {.} Linear fixed point: {\tt {find\_disp(m,a0)}} }{70}{subsection.1.A.2.1}
\contentsline {subsection}{\numberline {A.2.2}\color {.} Normalising the linear part: {\tt diag\_mat(m,a1,tune,damping) }}{70}{subsection.1.A.2.2}
\contentsline {subsection}{\numberline {A.2.3}\color {.} Normalising the nonlinear part: {\tt analyse\_kernel}}{72}{subsection.1.A.2.3}
\contentsline {subsubsection}{Phasors as eigenfunctions}{72}{section*.3}
\contentsline {subsubsection}{Lie maps and Lie operators}{73}{section*.4}
\contentsline {subsubsection}{The recursive algorithm of the normal form}{73}{section*.5}
\contentsline {section}{\numberline {A.3}\color {.} Canonization: {\tt \relax \fontsize {10.95}{13.6}\selectfont \abovedisplayskip 11\p@ plus3\p@ minus6\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6.5\p@ plus3.5\p@ minus3\p@ \belowdisplayskip \abovedisplayskip \let \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ CANONIZE(Atot,A\_cs,disp,a\_l,a\_nl,R,PHASE\_ADVANCE)}}{74}{section.1.A.3}
\contentsline {section}{\numberline {A.4}\color {.} Further explanations}{75}{section.1.A.4}
\contentsline {chapter}{Appendix \numberline {B}\color {.} Hierarchy of Analytical Methods}{77}{Appendix.1.B}
\contentsline {section}{\numberline {B.1}\color {.}Green's function Method}{77}{section.1.B.1}
\contentsline {subsection}{\numberline {B.1.1}\color {.}The rules of Analytical perturbation theory with maps}{78}{subsection.1.B.1.1}
\contentsline {subsection}{\numberline {B.1.2}\color {.}The Actual Calculation with Maps}{78}{subsection.1.B.1.2}
\contentsline {section}{\numberline {B.2} \color {.} Calculations with the Hamiltonian }{81}{section.1.B.2}
\contentsline {subsection}{\numberline {B.2.1} \color {.} Changing the time-like variable into a phase advance }{82}{subsection.1.B.2.1}
\contentsline {subsection}{\numberline {B.2.2}\color {.} The Fourier method approach: Guignard}{82}{subsection.1.B.2.2}
\contentsline {chapter}{Appendix \numberline {C}\color {.} De Moivre's formula and Invariant-Moments Duality}{87}{Appendix.1.C}
\contentsline {chapter}{Appendix \numberline {D} \color {.} The case of spin}{91}{Appendix.1.D}
\contentsline {section}{\numberline {D.1}\color {.} Spin Maps in FPP}{91}{section.1.D.1}
\contentsline {section}{\numberline {D.2}\color {.} Spin Maps Concatenation}{92}{section.1.D.2}
\contentsline {section}{\numberline {D.3}\color {.} Spin Normal Form}{92}{section.1.D.3}
\contentsline {section}{\numberline {D.4}\color {.} Factoring the Map $T$ and the invariant spin axis $\mathaccentV {vec}17E{n}$}{93}{section.1.D.4}
\contentsline {section}{\numberline {D.5}\color {.} A map with a single spin resonance}{95}{section.1.D.5}
\contentsline {chapter}{Appendix \numberline {E}\color {.} A Complex Package}{97}{Appendix.1.E}
\contentsline {section}{\numberline {E.1}\color {.} Presently implemented tools }{97}{section.1.E.1}
\contentsline {section}{\numberline {E.2}\color {.} Phasors Basis of this package}{99}{section.1.E.2}
\contentsline {section}{\numberline {E.3}\color {.} Phasors Basis: why do I reject symplectic phasors?}{101}{section.1.E.3}
\contentsline {section}{\numberline {E.4}\color {.}Some examples of tune/damping calculation }{103}{section.1.E.4}
\contentsline {section}{\numberline {E.5}\color {.}Stochastic beam envelopes: synchrotron Integrals }{104}{section.1.E.5}
\contentsline {section}{\numberline {E.6}\color {.} Spin calculation with the complex FPP}{107}{section.1.E.6}
\contentsline {section}{\numberline {E.7}\color {.} Stern-Gerlach Style Spin calculation with the complex FPP}{108}{section.1.E.7}
\contentsline {subsection}{\numberline {E.7.1}\color {.} Spin variables }{109}{subsection.1.E.7.1}
\contentsline {subsection}{\numberline {E.7.2}\color {.} A meat grinder algorithm applied for the spectator spin }{110}{subsection.1.E.7.2}
\contentsline {subsection}{\numberline {E.7.3}\color {.} The spin tune shift with amplitude }{111}{subsection.1.E.7.3}
\contentsline {subsection}{\numberline {E.7.4}\color {.} The spin invariant }{112}{subsection.1.E.7.4}
\contentsline {chapter}{Appendix \numberline {F}\color {.} Connection between MAD-X and PTC}{115}{Appendix.1.F}
\contentsline {section}{\numberline {F.1}\color {.} PTC part of {\tt als.h}}{115}{section.1.F.1}
\contentsline {section}{\numberline {F.2}\color {.} The actual program radiation.f90}{116}{section.1.F.2}
\contentsline {chapter}{Appendix \numberline {G}\color {.} PTC's new flat files}{117}{Appendix.1.G}
\contentsline {chapter}{Bibliography}{119}{chapter*.6}
